Finite Group Databasebeta

← #213653

Subgroups — #213653

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#9133C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#9131C212yesmaximal, complemented
7#9129C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8341C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#525—21no—
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8—21no—
16C22×C22⋊C8—21no—
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20#2856C2211yescharacteristic
21#2854—21no—
22#2854C412yes—
23#2850—21no—
24#2850—21no—
25#2847—21no—
26#2847C412yes—
27C4×#121—21no—
28C4×#121—21no—
29C4×#121C412yes—
30C4×#121C412yes—
31#2297—21no—
32#2285C2211yescharacteristic
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41C2×#336—21no—
42C2×#336—21no—
43C2×#332C2214yescomplemented
44C2×#331C2214yescomplemented
45#1296C2214yescomplemented
46#1296C2214yescomplemented
47C22×#41—21no—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53#532—44no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×#127C2311yescharacteristic
56C2×#124—21no—
57#525—44no—
58#525—44no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#120—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C412yes—
75C2×#117—21no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—44no—
92C2×C4⋊C8—44no—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2314yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—44no—
100C2×C22⋊C8—44no—
101#391—21no—
102#391—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390C2×C412yes—
108#390C2×C412yes—
109#389—21no—
110#389—21no—
111#389C2×C412yes—
112#389C2×C412yes—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C412yes—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C412yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#332—44no—
141#331—44no—
142C25C22⋊C411yescharacteristic
143C2×#41—44no—
144C2×#41—44no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4D814yes—
174C23×C4SD1614yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C4⋊Q8—24no—
193#127—24no—
194#124—44no—
195#123—48no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—42no—
205#121—42no—
206#121—42no—
207#121—42no—
208#121—44no—
209#121—42no—
210#121—42no—
211#120—48no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—48no—
215C4×D4—48no—
216#117—42no—
217#117—42no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271C4⋊C8—24no—
272C4⋊C8—44no—
273C4⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24#10214yes—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295#41—88no—
296#41—88no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×D4—24no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—416no—
379C22×C4—44no—
380C22×C4#10214yes—
381C22×C4#10214yes—
382C22×C4#10214yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×D812yes—
393C22×C4C2×D812yes—
394C22×C4C2×D812yes—
395C22×C4C2×SD1612yes—
396C22×C4C2×SD1612yes—
397C22×C4C2×SD1612yes—
398C22×C4#13514yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—88no—
411C2×C8—88no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42H(5,𝔽2)14yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—28no—
462C23—21no—
463C23—28no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23#41312yes—
482C23#41312yes—
483C23C2×#10212yes—
484C23C2×#10212yes—
485C23C2×#10212yes—
486C23#41612yes—
487C23#41712yes—
488C23C22×C22⋊C411yescharacteristic
489Q8—82no—
490Q8—82no—
491D4—84no—
492D4—88no—
493D4—88no—
494D4—88no—
495D4—82no—
496D4—82no—
497D4—82no—
498D4—82no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—88no—
557C2×C4C2×#10212yes—
558C2×C4C2×#10212yes—
559C2×C4C2×#10212yes—
560C2×C4#41612yes—
561C2×C4#41712yes—
562C2×C4#51812yes—
563C2×C4#53412yes—
564C2×C4#53412yes—
565C2×C4C22×D811yescharacteristic
566C2×C4C22×SD1611yescharacteristic
567C2×C4C2×#13512yes—
568C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
569C2×C4C2×#13711yescharacteristic
570C2×C4#57512yes—
571C8—84no—
572C8—84no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—416no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22#154014yes—
601C22#154714yes—
602C22#230811yescharacteristic
603C22C2×#41311yescharacteristic
604C22C2×#41311yescharacteristic
605C22C22×#10211yescharacteristic
606C22C2×#41611yescharacteristic
607C22C2×#41711yescharacteristic
608C22#289611yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#318612yes—
623C4#318712yes—
624C4#318812yes—
625C4#319012yes—
626C2—24no—
627C2—24no—
628C2—42no—
629C2—162no—
630C2C2×#154011yescharacteristic
631C2C2×#154711yescharacteristic
632C2#1771311yescharacteristic
633C2#1775311yescharacteristic
634C2#2175411yescharacteristic
635C2#2175811yescharacteristic
636C2#2176311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×#1797C2semidirect
4. #9133C2semidirect
5. #9133C2semidirect
6. #9131C2semidirect
7. #9129C2semidirect
8. #8726C2semidirect
9. #8341C2semidirect
43. C2×#332C22semidirect
44. C2×#331C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#366242
C2×#17972
#91332
#91312
#91292
#87262
#83412